The First Syzygy of Hibi Rings Associated with Planar distributive lattices

Abstract

Let $\mathcal{L}$ be a finite distributive lattice and $S=K[x_{\alpha }:\alpha \in \mathcal{L}]$ be a polynomial ring over a field $K$ and $I=⟨x_{\alpha }{x}_{\beta }-x_{\alpha \vee \beta }{x}_{\alpha \wedge \beta }:\alpha \nsim \beta ,\alpha ,\beta \in \mathcal{L}⟩$ an ideal of $S$. In this article we describe the first syzygy of the Hibi ring $R[\mathcal{L}]=S/I$, for a planar distributive lattice $\mathcal{L}$. We also derive an exact formula for the first Betti number of a planar distributive lattice. We give a characterization of planar distributive lattices for which the first syzygy is linear.

22 pages, 13 figures

References in corpus (2)

• Singular Loci of Hibi toric varieties
• Special functions associated to a certain fourth order differential equation